Video: Identifying the Coordinates of an Image of a Triangle after a Rotation with a Certain Degree given Its Verticesโ€™ Coordinates

Find the new positions of the given triangleโ€™s vertices after rotating it 180ยฐ counterclockwise about ๐ฟ.

04:06

Video Transcript

Find the new positions of the given triangleโ€™s vertices after rotating it 180 degrees counterclockwise about ๐ฟ.

You can imagine that weโ€™re going to put our finger on point ๐ฟ and spin the lines 180 degrees counterclockwise. Before we go any further, letโ€™s clarify what it means to go 180 degrees counterclockwise about a point, by looking at this point in yellow. If I want to take this point and turn this line 180 degrees counterclockwise, our yellow point would not change, but the line would go in the opposite direction.

Hereโ€™s what we should notice about our points. Point ๐ฟ will not change. The ๐ฟ prime or the new position of ๐ฟ will be the same as its old position, five, four. Now, letโ€™s look at ๐‘€ compared to ๐ฟ. To get to ๐‘€ from ๐ฟ, you go to the right three and up one. If we want a 180-degree counterclockwise position of ๐‘€, we need to do the opposite of this. We should go three places left and one place down. This will be our new ๐‘€, ๐‘€ prime.

We have to do the same process with our ๐‘. ๐‘ is located three units up and one unit to the right of ๐ฟ. To go 180 degrees counterclockwise about ๐ฟ, weโ€™ll need to do the opposite of three up and one to the right. Weโ€™ll go three units down and one unit left. This will give us the location of our new ๐‘, our ๐‘ prime.

Now, I can connect all of these points to see our new triangle. Hereโ€™s our new triangle. Again, you can imagine if you put your finger on the point at ๐ฟ, on that pink dot, you can see that this represents a 180-degree turn counterclockwise. We canโ€™t forget that this question wants to know the vertices of all three points. So we still need to identify where ๐‘€ prime and ๐‘ prime are on the graph.

What position is ๐‘€ prime in? Two units to the right and three units up. ๐‘€ prime is point two, three. ๐‘ prime is located at four units on the ๐‘ฅ-axis, four units to the right and one unit up. ๐‘ prime is found at point four, one. After we rotate our triangle 180 degrees counterclockwise about ๐ฟ, we have point ๐ฟ prime at five, four, ๐‘€ prime at two, three, and ๐‘ prime at four, one.

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